Question

Let M = ( (−3 1 3 4), (1 2 −1 −2), (−3 8 4 2))

14. (3 points) Let B1 be the basis for M you found by row reducing M and let B2 be the basis for M you found by row reducing M Transpose . Find the change of coordinate matrix from B2 to B1.

Answer #1

Let B1 = { u1, u2, u3 }, where u1 = (2,?1, 1), u2 = (1,?2, 1),
and u3 = (1,?1, 0). B1 is a basis for R^3 .
A. Find the transition matrix Q ^?1 from the standard basis of R
^3 to B1 .
B. Write U as a linear combination of the basis B1 .

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

. Given the matrix A =
1 1 3 -2
2 5 4 3
−1 2 1 3
(a) Find a basis for the row space of A
(b) Find a basis for the column space of A
(c) Find the nullity of A

Consider the following matrix.
A =
4
-1
-1
2
6
-3
6
4
1
Let B = adj(A). Find b31,
b32, and b33. (i.e., find
the entries in the third row of the adjoint of A.)

A=
2
-3
1
2
0
-1
1
4
5
Find the inverse of A using the method: [A | I ] → [ I | A-1 ].
Set up and then use a calculator (recommended). Express the
elements of A-1 as fractions if they are not already integers. (Use
Math -> Frac if needed.) (8 points)
Begin the LU factorization of A by determining a first
elementary matrix E1 and its inverse E1-1. Identify the associated
row operation. (That...

Let T(V)=AV be a linear transformation where A=(3 -2 6 -1 15, 4
3 8 10 -14, 2 -3 4 -4 20)
a.) construct a basis of the kernal T
b.) calculate the basis of the range of T
c.) determine the rank and nullity of T

Question 2: Let A = 2 −2 4 3 −2 5 −3 3 −4 . a.) Perform
elementary row operations to put A in echelon form. b.) Write A as
a product of a lower and upper triangular matrix, A = LU. c.)
Compute the determinant of L, U, and A.

Suppose B = {b1,b2} is basis for linear space V and C =
{⃗c1,⃗c2,⃗c3} is a basis for linear space W. Let T : V → W be a
linear trans- formation with the property that ⃗ T ( b 1 ) = 3 ⃗c 1
+ ⃗c 2 + 4 ⃗c 3 , ⃗ T ( b 2 ) = 4 ⃗c 1 + 2 ⃗c 2 − ⃗c 3 . Find the
matrix M for T relative to...

Let B = {(1, 3), (?2, ?2)} and B' = {(?12, 0), (?4, 4)} be bases
for R2, and
let A =
3
2
0
4
be the matrix for T: R2 ? R2 relative to B.
(a) Find the transition matrix P from B' to B. P =
(b) Use the matrices P and A to find [v]B and [T(v)]B, where
[v]B' = [1 ?5]T. [v]B = [T(v)]B =
(c) Find P?1 and A' (the matrix for T relative...

Let C = column matrix [1 2 3 1] and D = row matrix[−1 2 1 4] .
Prove that (CD)^10 is the same as C(DC)^9D and do the calculation
by hand.

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